Most on-site "deal with right away" stuff for me is over our professional level radio. This also prevents the general public from getting similar consumer FRS level radios and listening in.
Depending on the size of the event, it might be useful to have multiple channels, usually giving security its own, but with a feature for key staff members to do an all-call. It also helps to have a spare channel in case you need to have an extended conversation.
With radios, you can always slowly integrate it. First only do key staffers like department heads and then add on more as you grow in size and budget. Prioritizing is key there because not every staffer needs one and rental costs can escalate quickly.
It also helps to give staffers a quick training session on basic radio operations, lingo, and (I can't emphasize this enough) etiquette. No matter how stressed you get at con, yelling at someone over the radio is not acceptable.
Finally, always make sure you have plenty of spare batteries. During Anime Boston I have to swap mine out at least once a day.

-Doug Wilder
Resident Mecha fanboy of AnimeCons TV!

Edited by Nigoki on 07-09-14 10:07 PM. Reason for edit: No reason given.

If it's lower priority things, I tend to text message people to keep the radio channel free. You just have to make sure people are paying attention to their phones. Acknowledgement replies are very important.
Just make sure everyone had texting on their phone and unlimited texting on their phone plan. You can (and will) go through a lot of messages. Even if it's just for the month of the con, you need unlimited.